One computer implemented approach for calculating a demand forecast involves defining a so-called demand forecast tree capable of being graphically represented by a single top level node with at least two branches directly emanating therefrom, each branch having at least one further node. The demand forecast is computed on the basis of historical sales data typically associated with bottom level nodes of a demand forecast tree by a forecast engine capable of determining a mathematical simulation model for a demand process. One such forecast engine employing statistical seasonal causal time series models of count data is commercially available from Demantra Ltd, Israel, under the name Demantra™ Demand Planner.
Demand forecast applications include determining an optimal draw matrix D* specifying the number of copies of different consumer items to be delivered to each location of a multi-location single period inventory system to maximize the expected total profit (ETP) realizable by a distribution policy therefor in accordance with the expression:
                                                        ETP              =                            ⁢                                                ∑                                      i                    ⁢                                                                                  ⁢                    j                                                  ⁢                                  EP                                      i                    ⁢                                                                                  ⁢                    j                                                                                                                          =                            ⁢                                                ∑                                      i                    ⁢                                                                                  ⁢                    j                                                  ⁢                                  [                                                                                    (                                                                              p                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                -                                                      c                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                                      )                                            ⁢                                              D                                                  i                          ⁢                                                                                                          ⁢                          j                                                                                      -                                                                  (                                                                              p                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                -                                                      g                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                                      )                                            ⁢                                              ER                        ⁡                                                  (                                                                                    λ                                                              i                                ⁢                                                                                                                                  ⁢                                j                                                                                      ,                                                          D                                                              i                                ⁢                                                                                                                                  ⁢                                j                                                                                                              )                                                                                      -                                                                                                                                        ⁢                                                b                                      i                    ⁢                                                                                  ⁢                    j                                                  ⁢                                  EST                  ⁡                                      (                                                                  λ                                                  i                          ⁢                                                                                                          ⁢                          j                                                                    ,                                              D                                                  i                          ⁢                                                                                                          ⁢                          j                                                                                      )                                                              ]                                                          Eqn        .                                  ⁢                  (          1          )                    where pij is the unit retail price of an ith consumer item at a jth location, cij is its unit production cost, gij is its unit return cost when unsold, and bij is its unit stockout cost. Derived from Eqn. (1), the optimal draw matrix D* for the most profitable distribution policy for a single period inventory system is calculated using optimal availabilities Aij* where:
                              A                      i            ⁢                                                  ⁢            j                    *                =                              F            ⁡                          (                                                λ                                      i                    ⁢                                                                                  ⁢                    j                                                  ,                                  D                                      i                    ⁢                                                                                  ⁢                    j                                    *                                            )                                =                                                    p                                  i                  ⁢                                                                          ⁢                  j                                            -                              c                                  i                  ⁢                                                                          ⁢                  j                                            +                              b                                  i                  ⁢                                                                          ⁢                  j                                                                                    p                                  i                  ⁢                                                                          ⁢                  j                                            -                              g                                  i                  ⁢                                                                          ⁢                  j                                            +                              b                                  i                  ⁢                                                                          ⁢                  j                                                                                        Eqn        .                                  ⁢                  (          2          )                    
The unit retail price pij of an ith consumer item at a jth location, its unit production cost cij, and its unit return cost gij are tangible costs whilst its unit stockout cost bij reflects both tangible and intangible pecuniary considerations due to the lost sale of a unit such as loss of potential profit, customer goodwill, and the like. The unit stockout cost bij of an ith consumer item at a jth location of a single period inventory system is often estimated to be the unit profit pij−cij, however, this approach often renders an optimal availability Aij* incompatible with an industry accepted availability therefor denoted AijI, thereby implying that the unit stockout cost bij is, in fact, different than the unit profit pj−cj.
This is now exemplified for a single period inventory system for delivering newspapers for which typical unit cost values are as follows: pj=US$0.50, bj=cj=US$0.25, and gj=US$0.00 for all locations j. On the assumption that unit stockout cost bj equals unit profit pj−cj, this leads to an optimal common availability A*=66.7% using Eqn. (2). Against this, the industry accepted availability AijI for a single period inventory system for delivering newspaper is 80%±5%. The discrepancy between these availabilities implies that for the same jth location with mean demand λ=20, the draw Dj thereat would be 21 in accordance with the optimal common availability A*=66.7% and as high as 24 corresponding to AI=84.3%.